A company manufactures two models of MP3 players. Let x represent the number in millions of the
first model made, and let y represent the number in millions of the second model made.
The company's revenue can be modeled by the equation r of x, y.
We first want to find the marginal revenue equations, which would be the first order partial
derivatives. So to find the partial derivative of r with respect to x, we'll differentiate
with respect to x, treating y as a constant. The derivative of 140x with respect to x would be 140,
plus the derivative of 50y with respect to x would be zero since we're treating y as a constant,
minus the derivative of four x squared with respect to x. That would be minus eight x,
and then minus the derivative of two y squared with respect to x would be zero,
and then minus the derivative of xy with respect to x would be minus y.
Now for the partial derivative with respect to y, we'll differentiate with respect to y, treating x
as a constant. Therefore, the derivative of 140x with respect to y would be zero,
plus the derivative of 50y with respect to y would be 50,
minus the derivative of four x squared with respect to y would be zero,
minus the derivative of two y squared with respect to y would be
minus four y, and then minus the derivative of xy with respect to y would be minus x.
So here's the first part of our question. For the second part, we can achieve maximum revenue when
both partial derivatives are equal to zero. So we want to set our partial derivatives equal to zero
and solve as a system of equations to find the production levels that will maximize revenue.
Here are the partial derivatives from the first slide. Notice how I set them equal to zero,
and now we'll solve this as a system. To solve this system, let's use substitution.
Let's go ahead and solve this first equation here for y.
Notice if we add y to both sides of the equation, we could write this as y equals 140 minus eight x.
So now we'll substitute 140 minus eight x for y in the second equation and then solve for x.
Making this substitution, we would have 50
minus four times the quantity 140 minus 8x minus x equals zero.
Let's clear the parentheses and solve for x. So we'd have 50 minus 560 plus 32x
minus x equals zero. Combining like terms, we have negative 510 plus 31x equals zero.
Add 510 to both sides. That would give us 31x equals 510. Divide by 31.
So the exact value of x that would maximize revenue is 510 divided by 31.
Let's go and write that here. Let's also get a decimal approximation because remember this
represents the production level in millions of MP3 players. So 510 divided by 31,
if we round to four decimal places, this would be approximately 16.4516.
Notice in this form, we can tell that the production level is approximately 16.4516 million
iPods for model one. Then to find the production level for the second model, we'll determine y
where y is going to be equal to 140 minus eight times the exact value of 510 divided by 31.
So we'll get the exact value, and then we'll get a decimal approximation.
Here's the decimal approximation. If we press math, enter, enter,
we can get the exact value of 260 divided by 31.
Well, the decimal approximation to four decimal places would be approximately 8.3871.
So again, the exact value would be 260 divided by 31,
which is what we should enter. But to better understand this, it's approximately 8.3871.
And again, these are the production levels in millions of iPods. I hope you found this helpful.